Complex numbers extend the number system by adding the imaginary unit i, where i^2 = -1. A complex number is written as a + bi, with a real part and an imaginary part. These numbers make it possible to solve equations like x^2 + 1 = 0, which have no real solutions.
They also appear in physics, engineering, signal processing, and electrical circuits.
Key Facts
- Standard form: z = a + bi, where a is the real part and b is the imaginary part.
- Imaginary unit: i^2 = -1, so i = sqrt(-1).
- Addition: (a + bi) + (c + di) = (a + c) + (b + d)i.
- Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i.
- Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i.
- Division: (a + bi)/(c + di) = ((a + bi)(c - di))/(c^2 + d^2), where c + di is not 0.
Vocabulary
- Complex number
- A number that can be written in the form a + bi, where a and b are real numbers and i^2 = -1.
- Real part
- The value a in the complex number a + bi, shown on the horizontal axis of the complex plane.
- Imaginary part
- The value b in the complex number a + bi, shown as the coefficient of i on the vertical axis of the complex plane.
- Complex conjugate
- The number a - bi formed by changing the sign of the imaginary part of a + bi.
- Complex plane
- A coordinate plane where the horizontal axis represents real values and the vertical axis represents imaginary values.
Common Mistakes to Avoid
- Combining real and imaginary terms as if they are like terms is wrong because 3 and 4i are different types of parts and cannot become 7i or 7.
- Forgetting that i^2 = -1 is wrong because multiplying imaginary terms changes the sign, such as (2i)(5i) = 10i^2 = -10.
- Dividing complex numbers without using the conjugate is wrong because the denominator must be made real to write the answer in standard form.
- Using the wrong cycle for powers of i is wrong because powers repeat every 4: i, -1, -i, 1.
Practice Questions
- 1 Compute (4 + 3i) + (-2 + 7i) and write the answer in standard form.
- 2 Simplify (3 - 2i)(5 + 4i) and write the answer in standard form.
- 3 Explain why multiplying a complex number a + bi by its conjugate a - bi always gives a real number.